SeniorVBDStudent
Footballguy
Interesting developments...
All we really need is to calculate the average value of the distribution associated with max(QB1, QB2). This expectation, and the cost, can be directly compared to the elite QB.
Intuitively you would think this resolves to a simple formula, but that is only the case if the two variables have the same average and same standard deviation (references below).
In that simplified case the mean / expectation value of the max = average + (standard deviation)/sqrt (pi).
For the case we are interested in, where averages and deviations of the various players are not identical, you can use the procedure outlined earlier to solve P(max(QB1,QB2)) = 0.5 (probability is 50% around the mean). In excel you can use MATCH to find the score value which corresponds to this equality after computing all the probabilities in a table from 0 points to 100 points.
The implication is that you can create hybrid players as combinations of players, for any position and any number of players in that position (up to roster limits) and compute production and cost for each and every hybrid.
This can then be thrown into a linear optimization to find the unique global maximum for the best possible roster (based on assumed statistics). Excel may still limit the number of player variables to 200, so you may need an alternative freeware or payware to perform the optimization.
QED.
References associated with expectation value of max(QB1, QB2) (with and without assumption of identical averages and standard devlation):
https://math.stackexchange.com/questions/1804353/expected-values-of-maxx-y-and-minx-y-for-n-mu-sigma2-distributed (proves the simple formula)
https://mathoverflow.net/questions/298920/mean-and-variance-of-maximum-of-random-variables (explains why the formula is not simple for non identical distributions)
All we really need is to calculate the average value of the distribution associated with max(QB1, QB2). This expectation, and the cost, can be directly compared to the elite QB.
Intuitively you would think this resolves to a simple formula, but that is only the case if the two variables have the same average and same standard deviation (references below).
In that simplified case the mean / expectation value of the max = average + (standard deviation)/sqrt (pi).
For the case we are interested in, where averages and deviations of the various players are not identical, you can use the procedure outlined earlier to solve P(max(QB1,QB2)) = 0.5 (probability is 50% around the mean). In excel you can use MATCH to find the score value which corresponds to this equality after computing all the probabilities in a table from 0 points to 100 points.
The implication is that you can create hybrid players as combinations of players, for any position and any number of players in that position (up to roster limits) and compute production and cost for each and every hybrid.
This can then be thrown into a linear optimization to find the unique global maximum for the best possible roster (based on assumed statistics). Excel may still limit the number of player variables to 200, so you may need an alternative freeware or payware to perform the optimization.
QED.
References associated with expectation value of max(QB1, QB2) (with and without assumption of identical averages and standard devlation):
https://math.stackexchange.com/questions/1804353/expected-values-of-maxx-y-and-minx-y-for-n-mu-sigma2-distributed (proves the simple formula)
https://mathoverflow.net/questions/298920/mean-and-variance-of-maximum-of-random-variables (explains why the formula is not simple for non identical distributions)
